Numerical Analysis Ii

1. Apply polynomial approximation techniques to estimate function values.
2. Utilize orthogonal polynomials for function approximation and data fitting.
3. Implement cubic spline and Hermite interpolation methods.
4. Apply numerical integration methods to approximate definite integrals.
5. Solve boundary value problems using finite difference schemes.
6. Classify and solve partial differential equations numerically.

13 weeks, about 87 hours in total. Yours will differ.

1. Week 1Module 1: Approximations

   • Unit 1: Polynomials · 4 hours

     Understand the definition of a polynomial and its degree.. Distinguish between polynomial functions and polynomial equations.. Express simple functions as polynomials using series expansion.. Identify different types of function approximation methods..

   • Unit 2: Least Squares Approximation (Discrete Case) · 5 hours

     Understand the basic idea of least square approximation.. Derive the least square formula for discrete data.. Learn to fit a linear polynomial to a set of data points.. Learn to fit a quadratic or parabolic polynomial to a set of data points..

2. Week 2Module 1: Approximations

   • Unit 3: Least Squares Approximation (Continuous Case) · 6 hours

     Distinguish between discrete data and continuous functions.. Learn to fit polynomials to continuous functions using the least squares approach.. Practice integration techniques required for continuous least squares approximation..

3. Week 3Module 2: Orthogonal Polynomials

   • Unit 1: Introduction to Orthogonal System · 5 hours

     Define orthogonal polynomials and understand their properties.. Formulate orthogonal and orthonormal polynomials.. Handle inner product of functions and understand its properties.. Verify orthogonality of functions through integration..

4. Week 4Module 2: Orthogonal Polynomials

   • Unit 2: The Legendre Polynomials · 5 hours

     State the Rodrigues' formula for generating Legendre polynomials.. Generate Legendre polynomials using Rodrigues' formula and recurrence relation.. Understand the orthogonality property of Legendre polynomials.. Solve problems using Legendre polynomials.

5. Week 5Module 2: Orthogonal Polynomials

   • Unit 3: Least Squares Approximation by Legendre Polynomials · 5 hours

     Apply Legendre polynomials to least squares procedures.. Obtain least square approximations using Legendre polynomials.. Solve numerical problems using Legendre polynomial approximation..

6. Week 6Module 2: Orthogonal Polynomials

   • Unit 4: The Chebyshev Polynomials · 5 hours

     State the necessary formulae for generating Chebyshev polynomials.. Obtain Chebyshev polynomials Tn(x) up to n = 10 using recurrence formula.. Classify Chebyshev polynomials as a family of orthogonal series.. Understand the properties of Chebyshev polynomials..

7. Week 7Module 2: Orthogonal Polynomials

   • Unit 5: Series of Chebyshev Polynomials · 5 hours

     Identify the form of functions suitable for Chebyshev polynomial approximation.. Apply Chebyshev polynomials to fit a cubic approximation to a function f(x).. Evaluate the accuracy of Chebyshev polynomial approximations..

8. Week 8Module 2: Orthogonal Polynomials

   • Unit 6: Chebyshev Interpolation · 5 hours

     Use Lagrange's formula for interpolation.. Interpolate using Chebyshev polynomials.. Compute the error table from the approximation.. Apply Chebyshev interpolation to approximate functions..

9. Week 9Module 3: Further Interpolation Techniques

   • Unit 1: Cubic Spline Interpolation · 6 hours

     Define a cubic spline and understand its properties.. Derive a method of fitting a cubic spline to a set of data points.. Fit a cubic spline to a set of data points.. Interpolate a function from the fitted cubic spline..

10. Week 10Module 3: Further Interpolation Techniques

    • Unit 2: Hermite Approximations · 5 hours

      Distinguish between cubic spline and Hermite polynomial.. Figure out the Hermite approximation formula.. Fit polynomial by Hermite approximation technique.. Find an estimate using Hermite approximation..

11. Week 11Module 4: Numerical Integration

    • Unit 1: Introduction to Numerical Integration · 4 hours

      Understand the concept of numerical integration.. Differentiate between analytical and numerical approaches to integration.. List various known methods for numerical quadrature.. Understand the role of polynomial approximation in numerical integration..

    • Unit 2: Trapezoidal Rule · 5 hours

      Derive the Trapezoidal rule geometrically.. Use the Newton-Gregory formula to derive the Trapezoidal rule.. Implement the Trapezoidal rule to evaluate a definite integral.. Estimate the error in the Trapezoidal rule..

12. Week 12Module 4: Numerical Integration

    • Unit 3: Simpson's Rules · 5 hours

      Derive Simpson's 1/3 rule using Newton Forward formula.. Distinguish between Simpson's rule and Trapezoidal rule.. Apply Simpson's 1/3 rule to evaluate definite integrals.. Understand the conditions for applying Simpson's rule..

    • Unit 4: Newton-Cotes Formulas · 5 hours

      Derive the Simpson's 3/8 rule.. Understand the structure of Newton-Cotes formulas.. Apply Newton-Cotes formulas to evaluate definite integrals.. Compare the accuracy of different Newton-Cotes formulas..

13. Week 13Module 5: Boundary Value Problems

    • Unit 1: Introduction to BVP · 6 hours

      Distinguish between Initial Value Problems and Boundary Value Problems.. Derive finite difference schemes for solving BVPs.. Solve BVPs using finite difference schemes.. Understand the application of Taylor series in deriving finite difference approximations..

    • Unit 2: BVP involving Partial Differential Equation · 6 hours

      Define a second-order PDE and a BVP involving a PDE.. Classify various types of PDEs (parabolic, elliptic, hyperbolic).. Classify types of boundary conditions for PDEs (Dirichlet, Neumann, Mixed).. Derive finite difference schemes for PDEs..